Internat. J. Math. & Math. Sci.
VOL. 21 NO.
(1998) 153-158
153
SOME RESULTS ON COMPACT CONVERGENCE SEMIGROUPS
DEFINED BY FILTERS
PHOEBE HO, PAUL PLUMMER, and SHING SO
Central Missouri State University
Warrensburg, Missouri
(Received June i0, 1996 and in revised form September 30, 1996)
ABSTRACT. In this paper the concept of convergence defined by filters is used and applied
in the study of semigroups. Special emphasis is placed on compact convergence semigroups and
their properties.
KEY WORDS AND PHRASES: convergence semigroups, convergence groups, maximal
subgroups, compact, pseudotopological, dense.
AMS(MOS) SUBJ. CLASSIFICATION CODES: Primary: 54A20, 54Hll, 22M99
1. INTRODUCTION.
Let
S which contains {x}. Then is an
ultra.filter called the principle ulrafilfer In [1], Kent’s approach to convergence was to set up
a mapping q from F(X), the set of all filters on a set X, to P(X), the power ,c* of X. Then a
filter
on X is said to q converge to x in X, denoted by 9r
x, if x 6 q(’).
DEFINITION. A convergence space (X, q) is a non-empty set
a mapping q bctween
/ be the set of all subsets of a non-empty set
"
F(X)
and
X’and
P(X)
which satisfy the following conditions:
i) x, forallx6X;
ii) if ’-xand ’<_G, then Gx;
iii) if ’xand x, then’nGx.
When these properties are satisfied q is known as a convergence structure. If the convergence structure is fixed for a specific discussion, as in reference to
we will refer to
a
general convergence space,
-
(X, q) as "the convergence space X". A convergence space X is Hausdorff if each
-
filter converges to at most one point and X is compa..ct if every ultrafilter converges in X.
is a
Y. If
DEFINITION. Let (X, q) and (Y, p) be convergence spaces and f" X
filter on S, then f(.T’) will denote the filter on Y with f(F) F
.T" as its base. The function
f is said to be continuous at x if f(’) p-converges to f(x) whenever
q-converges to x.
DEFINITION. A closure operation on a convergence space S is defined in the following
x.
and
such that A
way. If A C S and x S, then x A if there exists a filter
"
" "
154
P. HO, P. PLUMMER AND S. SO
A is defined to be closed if and only if A
A.
The following five lemmas are immediate results from the definitions involved.
LEMMA 1.1. If A is a compact subset of X and f: (X, q)
(Y, p) is a continuous
function, then f(A) is compact.
LEMMA 1.2. If X is a compact convergence space and T a closed subset of X, then T is
compact.
LEMMA 1.3. If X is a compact space, and 2) is a descending family of non-empty closed
subsets of X, then N:D }.
LEMMA 1.4. If X is a Hausdorff convergence space and T a compact subset of X, then
T is closed.
LEMMA 1.5. If A and B are compact subsets of a convergence space X, then A xB is
compact in the product convergence structure on XxX.
2.
MAIN RESULTS.
DEFINITION. A convergence semigroup is
tinuous function m: SS
S such that:
a convergence space
S together with
a con-
i) S is Hausdor;
ii) m is associative.
The following notations are useful in the discussion of convergence semigroups:
i) For a, b S, ab m((a, b)).
ii) For A, B C_ S, AB rn(AxB)
{ablaAandbB}. In particular, A{D}
will be
denoted Ab.
iii) .TxisthefilteronSxSwith{FGIFUandG} as its base.
iv) .T’. is the filter on S with rn(.T’{) as its base.
v) ’a is the filter with {Fa If Y} as its base.
LEMMA 2.1. If and are filters on a convergence semgroup S such that .T" x
and
y, then .T. { xy.
LEMMA 2.2. H A and B are compact subsets of a convergence semigroup S, then AB is
compact in S.
DEFINITION. If f: X X, then {x X If(x x} is called the set of fixed points of
"
The next lemma shows two properties of fixed points in convergence spaces.
LEMMA 2.3. Let X be a Hausdroff convergence space and F the set of fixed points of a
X. Then the following is true:
continuous function f: X
r
i) If F 9 a filter on X, then f(-) ;
ii) If X is Hausdorff, then F is closed.
PROOF. i) IrA f(’), then there existsA* ’suchthat f(A*) C_ A. But F 9v
and FNA* .T’. Now FA* _C F implies that F A*
implies that FNA* :fi
/(FOA*) _C
RESULTS ON COMPACT CONVERGENCE SEMIGROUPS
-
155
f(A*) c_ A which implies that A E.. Therefore, f() C_ ’.
and f(ANF) ANF. This implies that ANF
If A E ’, then ANF
f("). Now ANF
C_ A implies that A
f(’). Therefore, v C_ f(’).
y. Since f is
ii) Let y F. Then there exists afilter.T such that F E and
f(y), and from part i), f(Y’) Y’, so Y" f(y). But X being
continuous, we know f(Y)
F.
Hausdorff implies that f(y) y. So y F. Therefore, F C_ F so F
DEFINITION. An element e of a semigroup S is called an idempotent if e
e. We
use E(S) to denote the set of all idempotents in S.
-
"
DEFINITION. A nonempty subset T of a semigroup S is said to be a subsemigroup
T C_ T. A nonempty subset G of S is a subgroup of S if G is a group under the
if T.T
multiplication defined on S.
THEOREM 2.1. If
a
Hausdorff convergence semigroup S is compact
then S contains
an idempotent.
PROOF. We will show that S contains a minimal closed subsemigroup and that every such
.
subsemigroup consists of a single idempotent. Let S denote the set of closed subsemigroups of
S. Note S S, so ,5’ # Partially order S by inclusion and let be a maximal tower in S by
use of the Hausdorff Maximality Principal.
Let T
fXJ. Then, from Lemma 1.3, T
# 0. Let
T. Since T is a non-empty closed subsemigroup of S, Lemma 1.2 implies that T is compact;
hence, by Lemma 2.2, xT is compact. Therefore by Lemma 1.4, xT is a closed subsemigroup
of S contained in T. Now, by the maximality of C, we see that xT T and, similarly, Tx T.
x
Thus T is a subgroup of S. If e is the identity of T, the maximality of C ensures that T
Let f be
"
r,(f(F))
and let
mapping from from the semigroup S into S x S by f(x)
be a filter on S such that
x. If F
:, the f(F)
F for
1, 2,
composite function
S by
f o rn" S
(x, x) for all
a
f o rn.
so
Since
f o re(x)
"
f(’) (x, x).
f
Therefore
f
x 2. Then, by
{(,x)
is continuous.
and rn are both continuous, we know
{e}.
x
e S,
F}. Now
Next, consider the
f o rn is continuous
and
Lemma 2.2 the set of fixed oints of f o rn is closed.
But the set of fixed points of f o m is E(S), the set of idempotents in the semigroup. Therefore,
we have the following theorem.
THEOREM 2.2. The set E(S), of all idempotents of a convergence semigroup S, is closed.
THEOREM 2.3. Let S be a convergence semigroup and G a subgroup of S. Then G is a
subsemigroup of S with identity.
PROOF. Let x, y E G. This implies that there exists
"
and (; such that G is contained
-
inY’and,andY’x,--,y. Since G is a group, G.G=GsoGffY-and’---,xy.
Therefore, xy
G
so
C_ G.
Now if e is the identity of G and x
But since 6
Let F
e, we know 6-Y"
’. Then G,F
ex.
G, then there exists Y such that G Y" and
Also since {e} E 6 and eG G, we have G eG
’which implies that GNF
x.
’and since GFC_G we know
e(GNF)=GCFsoGflFE6..’. But Gr3FCFimpliesF6.’whichmeans
P. HO, P. PLUMMER AND S. SO
156
So
."
x.
Now S being Hausdorff implies that
ex
x. Therefore, e is the
x, similarly xe
identity for G.
THEOREM 2.4. If S is
G is also
a compact convergence semigroup and
a
subgroup of S, then
- .
subgroup.
a
G
-
PROOF. By Theorem 2.3, it suffices to show if x E G then x has an inverse which is
e. Let y
Now if y G.
equivalent to the existence of an x -1 E such that x-ix xx
G, then there exists a filter such
that G E .T" and .T"
y. For each F C_ G, let F
{x Ix E F} and ’-1 the filter with
F- F E Y and F C_ G} as its base. Now there exists an ultra.filter such that --1 < and,
and G -1 G, G ’-1 so G
since S is compact, 7-/--, h for some h S. Now since G
_< H and
.’. Since
and hE. By Lemma 2.1, ."
hy so consider HF
FfG e ’, (FNG) -1 e/ which implies HN(FNG) 7-/ and HN(FG) -1#(3. Thus
such that x -IH and x(FNG) C_F so x-x=eHF, so
there exists x E(FG)
which implies "H..T" _<
e HF for all HF E ?’/.
hy and we know
Therefore,
but the fact that S is Hausdorff implies that hy e. Similarly, we can show yh e. Thus,
h=y -1 G.
G is
a group so y has an inverse and we are done. If y
-1
-1
"
-.
- "-
DEFINITION. Let G be
f" G
a
-
subgroup of
G defined by f(x)
semigroup.
x
’-
a convergence semigroup
S. If the inversion
is continuous, then G is called a convergence
DEFINITION. A convergence space X is said to be pseudotopological if a filter
converges to some point x in X if and only if all ultraalters finer than
converge to x.
"
THEOREM 2.5. If S is a compact pseudotopological semigroup, which is algebraically
a group, then S is a convergence group.
PROOF. Let f" S
"
S defined by f(x)
x
-
and
is a filter in S. Let ?’f be a ultrafilter such that T/>
some y in
S. Since for every H
and F
that xy
?’/, HN f(
9r
Similarly, yx
e. Thus y
z
-
a filter such
-
-#- } for all
where e is the identity of the group S. Now
e.
"
f( ).
"
F
."
x. Then
f(
Since S is compact, ?-/ y for
T/ y,
Then e
?’/<
Since S is pseudotopological,
continuous. Therefore, S is a convergence group.
"
that
f(
HF for all H E
,
"
and
e
z
-
DEFINITION. A subset D of a convergence space X is said to be dense in X if D
imply
so f is
X.
DEFINITION. If S is a convergence semigroup and e E(S), then the largest proper
subgroup of S which contains e is called the maximal subgroup of S containing e.
THEOREM 2.6. i) If S is a compact convergence semigroup, then each maximal subgroup
is either closed or dense in S.
ii) If S is
a
compact pseudotopological semigroup, then either every maximal subgroup of S
is closed or S is a convergence group.
iii)
If S is a compact pseudotopological semigroup, then every maximal subgroup of S is a
RESULTS ON COMPACT CONVERGENCE SEMIGROUPS
157
.
convergence group.
PROOF. i) By Theorem 2.4, if G is a subgroup then so is
SinceGCCS,isa
group, and G is maximal, either G G and G is closed, or G S and G is dense.
ii) If S does have a dense subgroup, then G S implies S is a group. Since S
pseudotopological, S is a convergence semigroup by Theorem 2.5.
is compact and
iii) Let G be a maximal subgroup of S. Then G is either closed or dense in S.
If G is closed, then G satisfies all the hypothesis of Theorem 2.5. so G is
a convergence
group.
If G is dense in S, then by ii) S is a convergence group. Since the inversion mapping on G is
a restriction of the inversion mapping on
S, it must be continuous. Therefore G is a convergence
group.
REFERENCES
1.
KENT, D.C., Convergence Functions and Related Topologies, Fundamenta Mathematicae
54 (1964), 125-133.
2.
WILLARD, S., General Topology, Addison-Wesley, Reading, Massachusetts, 1970.
3.
PAALMAN-DE MIRANDA, A.B., Topological Semigroups, Mathematical Centre Tracts,
2nd Ed., Mathematiche Centrum, Amsterdam, 1970.
4.
CARRUTH, J.H., HILBEBRANT, J.A., and KOCH, R.J., The Theory of Topological Semigroups, Marcel Dekke, Inc., New York, 1983.
5.
HUSAIN, T., Introduction to Topological Groups, W. B. Saunders Company, Philadelphia
and London, 1966.
6.
PLUMMER, P.J., Convergence Semigroups in Convergence Spaces Defined by Filters, Thesis, Central Missouri State University, 1995.